Question: The lifespans of tigers in a particular zoo are normally distributed. The average tiger lives $18.9$ years; the standard deviation is $3.6$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a tiger living longer than $11.7$ years.
Solution: $18.9$ $15.3$ $22.5$ $11.7$ $26.1$ $8.1$ $29.7$ $95\%$ $2.5\%$ $2.5\%$ We know the lifespans are normally distributed with an average lifespan of $18.9$ years. We know the standard deviation is $3.6$ years, so one standard deviation below the mean is $15.3$ years and one standard deviation above the mean is $22.5$ years. Two standard deviations below the mean is $11.7$ years and two standard deviations above the mean is $26.1$ years. Three standard deviations below the mean is $8.1$ years and three standard deviations above the mean is $29.7$ years. We are interested in the probability of a tiger living longer than $11.7$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $95\%$ of the tigers will have lifespans within 2 standard deviations of the average lifespan. The remaining $5\%$ of the tigers will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({2.5\%})$ will live less than $11.7$ years and the other half $({2.5\%})$ will live longer than $26.1$ years. The probability of a particular tiger living longer than $11.7$ years is ${95\%} + {2.5\%}$, or $97.5\%$.